Effect of Anisotropic Permeability on Thermosolutal ...

International Journal of Fluid Mechanics & Thermal Sciences 2020; 6(4): 124-131

http://www.sciencepublishinggroup.com/j/ijfmts

doi: 10.11648/j.ijfmts.20200604.13

ISSN: 2469-8105 (Print); ISSN: 2469-8113 (Online)

Review Ariticle

Effect of Anisotropic Permeability on Thermosolutal Convection in a Porous Cavity Saturated by a Non-newtonian Fluid

Yovogan Julien1, *

, Fagbemi Latif2, Koube Bocco Sèlidji Marius

3, Kouke Dieudonné

2,

Degan Gérard1

1Department of the Energizing Genius and Environment, National University of Sciences, Technologies, Engineering and Mathematics

(UNSTIM), Abomey, Republic of Benin 2Department of Renewable Energy and Energizing System, University of Abomey Calavi, Abomey Calavi, Republic of Benin 3Department of Physics, National University of Sciences, Technologies, Engineering and Mathematics (UNSTIM), Abomey, Republic of

Benin

Email address:

*Corresponding author

To cite this article: Yovogan Julien, Fagbemi Latif, Koube Bocco Sèlidji Marius, Kouke Dieudonné, Degan Gérard. Effect of Anisotropic Permeability on

Thermosolutal Convection in a Porous Cavity Saturated by a Non-newtonian Fluid. International Journal of Fluid Mechanics & Thermal

Sciences. Vol. 6, No. 4, 2020, pp.124-131. doi: 10.11648/j.ijfmts.20200604.13

Received: November 13, 2020; Accepted: November 30, 2020; Published: December 16, 2020

Abstract: In this paper, an analytical study is reported on double-diffusive natural convection in a shallow porous cavity

saturated with a non-Newtonian fluid by using the Darcy model with the Boussinesq approximations. A Cartesian coordinate

system is chosen with the x- and y- axes at the geometrical center of the cavity and the y’-axis vertically upward. The top and

bottom horizontal boundaries are subject to constant heat (q) and mass (j) fluxes. The porous medium is anisotropic in

permeability whose principal axes are oriented in a direction that is oblique to the gravity vector. The permeabilities along the

two principal axes of the porous matrix are denoted by K1 and K2. The anisotropy of the porous layer is characterized by the

permeability ratio K*=K1/K2 and the orientation angle φ, defined as the angle between the horizontal direction and the

principal axis with the permeability K2. The viscous dissipations are negligible. Based on parallel flow approximation theory,

the problem is solved analytically, in the limit of a thin layer and documented the effects of the physical parameters describing

this investigation. Solutions for the flow fields, Nusselt and Sherwood numbers are obtained explicitly in terms of the

governing parameters of the problem.

Keywords: Heat Transfer, Mass Transfer, Isotropy, Anisotropy

1. Introduction

Double-diffusive, or thermosolutal, natural convection is a

fluid motion due to simultaneous variations of temperature

and concentration in the gravity field. Thermodiffusion in

different fluids is a subject of intensive research due to its

wide range of applications in many engineering and

technological areas, including chemical engineering

(deposition of thin films, roll-over in storage tanks containing

liquefied natural gas, solution mining of salt caverns for crude

oil storage), solid-state physics (solidification of binary alloy

and crystal growth), oceanography (melting and cooling near

ice surfaces, sea water intrusion into freshwater lakes and the

formation of layered or columnar structures during

crystallization of igneous intrusions in the Earth’s crust),

geo-physics (dispersion of dissolvent materials or particulate

matter in flows), etc. Non-Newtonian flows are of importance

and very present in many industrial processes such as paper

International Journal of Fluid Mechanics & Thermal Sciences 2020; 6(4): 124-131 125

making, oil drilling, slurry transporting, food processing,

polymer engineering and many others. Some of these

processes are discussed by Jaluria [1].

There are very few investigations dealing with

double-diffusion convection inside rectangular enclosures

confining non-Newtonian fluids. Among them are those

performed while considering a saturated porous medium [2, 3].

In the case of a clear fluid medium, the only one is that

conducted lately by Makayssi et al. [4]. All these studies have

examined analytically and numerically the effect of the flow

behavior index, the Lewis number and the buoyancy ratio on

convection heat and mass transfers in the situation where both

thermal and solutal buoyancy driven forces act in the same

direction (the aiding case)..

Natural convection heat and mass transfer along a vertical

plate embedded in a power-law fluid saturated Darcy porous

medium in presence of the Soret and Dufour effects is

considered by Srinivasacharya and Swamy [5]. It can be

concluded according to their analysis that increasing the Soret

number (or decreasing the Dufour number) decreases the

Nusselt number, but increases the Sherwood number for shear

thinning, Newtonian and shear thickening fluids.

The influence of nanoparticles on natural convection

boundary layer flow inside a square cavity with

water-Al2O3nanofluid is accounted by Salma et al. [6].

Various Soret-Dufour coeffcients and Schmidt number have

been considered for the flow, temperature and concentration

fields as well as the heat and mass transfer rate, horizontal and

vertical velocities at the middle height of the enclosure while

Pr and Ra are fixed at 6.2, 104 respectively. The results of the

numerical analysis lead to the following conclusions: the

structure of the fluid streamlines, isotherms and

iso-concentrations within the chamber is found to significantly

depend upon the Soret-Dufour coeffcients.

Numerical simulation to investigate the effect of heat

generation, thermal Rayleigh number, solid volume fraction

and Lewis number on flow pattern, temperature field and

concentration in a triangular cavity filled porous media

saturated Al2O3 water nanofluid is conducted by Raju et al.

[7]. They found that the heat generation plays an important

role on fluid flow pattern, heat and mass transfer and the flow

strength is increased for rising the value of heat generation

parameter but decrease for increasing values of Lewis

number.

Double-diffusive natural convection in a porous arc-shaped

enclosure has been numerically studied by Ariyan et al. [8]

using Darcy-Brinkman formulation. The impacts of Darcy

number, Rayleigh number, Buoyancy ratio, and Lewis number

on flow pattern and heat and mass transfer process have been

investigated. It was shown for aiding flow (N < 0) the strength

of the flow is greater than that of opposing flow (N > 0). It

leads to greater heat and mass transfers for aiding flow case.

Any increment in the Darcy number varies heat and mass

profiles visibly, as isotherms and iso-concentrations are

distributed in the bulk of the cavity for higher Darcy numbers.

An increment in the Lewis number results in an

improvement/deterioration of mass/heat transfer process.

Unsteady double-diffusive natural convection flow in an

inclined rectangular enclosure subject to an applied magnetic

field and heat generation parameter is studied by Sabyasachi

and Precious [9]. They obtained the average Nusselt numbers

and average Sherwood numbers for various values of

buoyancy ratio and different angles of the magnetic field by

considering three different inclination angles of the enclosure

while keeping the aspect ratio fixed. Their results indicate that

the flow pattern, temperature and concentration fields are

significantly dependent on the buoyancy ratio and the

magnetic field angles.

The problem of natural convection in a horizontal fluid

layer subject to horizontal gradients of temperature and solute

has been solved by both analytical and numerical methods

[10]. The influence of the thermal Rayleigh number RaT, and

parameter a (=0, double diffusive convection; and=1 Soret

induced convection) on the intensity of convection were

predicted and discussed.

The effects of internal heat generation (IHG), combined

effects of Soret and Dufour with variable fluid properties like

variable porosity, permeability and viscosity on thermal and

diffusion mixed (free and forced) convection flow fluid

through porous media over moving surface is analyzed by

Girinath and Dinesh [11]. They found that the velocity and

temperature profiles decreased for an increase in prandtl

number Prwhere as the concentration profile is increased for

an increase prandtl number Pr.

Double-diffusive natural convection flow in a trapezoidal

cavity with various aspect ratios in the presence of

water-based nanofluid and applied magnetic field in the

direction perpendicular to the bottom and top parallel walls is

studied by Mahapatra et al. [12]. Their results indicated that

the strength of vortex decreases/increases as the magnetic field

parameter/aspect ratio increases. It was also found that

increase in the Rayleigh number causes natural convection

due to the increase in the buoyancy forces. In nanofluid, mass

transfer ratio was more effective than base fluid.

Authors who drove studies in this domain, made the

hypothesis that the porous medium is isotrope and saturated

by a newtonian fluid/ / non-newtonian fluid. Anisotropy,

which is generally a consequence of a preferential orientation

or asymmetric geometry of the grain or fibres, is in fact

encountered in all those applications in industry and nature.

[13] and others, studied double-Diffusive Natural Convection

with Cross-Diffusion Effects in an Anisotropic Porous

Enclosure Using ISPH Method, but they considered that the

porous medium is saturated by a newtonian fluid. In the

present paper we consider that the porous medium (saturated

by a non-newtonian fluid) is homogeneous and anisotropic in

permeability with arbitrarily oriented principal axes, as seen in

nature and many practical applications. An analytical model,

based on the parallel flow approximation, is proposed for the

case of a shallow layer (A >> 1), to determine the effect of

anisotropic parameters of the porous matrix on the speed field,

the temperature field, the concentration field and the heat

transfer.

126 Yovogan Julien et al.: Effect of Anisotropic Permeability on Thermosolutal Convection in a Porous Cavity

Saturated by a Non-newtonian Fluid

2. Mathematical Formulation

Figure 1. Physical model and coordinate system.

The physical system consists of a shallow rectangular

cavity field with a porous medium characterized by an

anisotropic permeability. The enclosure is of height H and

horizontal length L. The generated out-flow is laminaire. The

transfer of heat by radiance is negligible. The fluid is binary,

non-newtonian and incompressible.

A Cartesian coordinate system is chosen with the x- and y-

axes at the geometrical center of the cavity and the y’-axis

vertically upward. The top and bottom horizontal boundaries

are subject to constant heat (q) and mass (j) fluxes. The porous

medium is anisotropic, the permeabilities along the two

principal axes of the porous matrix are denoted by K1 and K2.

The anisotropy of the porous layer is characterized by the

permeability ratio K*=K1/K2 and the orientation angle φ,

defined as the angle between the horizontal direction and the

principal axis with the permeability K2. The continuity,

momentum, energy and concentration equations for the

porous cavity field are

* Equation governing the conservation of mass:

�� · ��’ � 0 (1)

*Equation governing the conservation of momentum

(modified Darcy model proposed by ([15, 16])).

��’ � �� ’� ��’ � �’�� (2)

*Equation governing the conservation of energy.

�’ ��’��’ � ��. ��’ � ��’ (3)

*Equation governing the conservation of concentration.

�’ ��’��’ � ��. ��’ � ��’ (4)

*Boussinesq equation.

�’ � � !1 #�’ �’ $ #�’ �’ $% (5)

In these equations, ��’ denotes the velocity vector, p′ the

pressure, T ′ the temperature and S′ the concentration.

Moreover, µ′a the apparent viscosity, g the gravitational

acceleration, �’ and �’ the constant reference Kelvin

temperature and concentration, αS the mass diffusivity in

porous medium, � the density of the fluid at �’ , & ′ the

density, βT the thermal-expansion coefficient, βS the

concentration-expansion coefficient, αT the thermal diffusivity,

ε′ the porosity of the porous medium,. In Eq. (2), the

symmetrical second-order permeability tensor '� is defined as

'� � (')*+,�- � '�./*�- #'� ')$ sin - cos -#'� ')$ sin - cos - '�*+,�- � ')./*�- 5 (6)

The boundary conditions of the system are as follows:

6’ � 7 8’� 9’ � 0, ��’�;’ � 0, ��’�;’ � 0 (7)

<’ � 7 =’� 9’ � 0, ��’�;’ � >?@ , ��’�;’ � A’B (8)

where k is the thermal conductivity and ψ′ the stream function

Then, the dimensionless formulation (ψ, T, S) of governing

equations become:

C �DE�FD � 2. �DE�;�F � H �DE�;D � ) � �I#6, <$ � J#6, <$�I#6, <$ � KC �#�LM�$�; � C �E�F � ��F � . �E�; � ��FJ#6, <$ � . �E�F � ��; � H �E�; � ��; NOP

OQ (9)

�E�F ��; �E�; ��F � �� (10)

�E�F ��; �E�; ��F � )8R �� (11)

where Ra=(ρ0.g.K1.∆T.βT. (H/αT)n)/ν the modified thermal

Rayleigh number, N=βS..∆S/βT.∆T the buoyancy ratio, Le=αT

/αS the Lewis number and ψ is the usual stream function

defined as:

S � �E�F , T � �E�; (12)

such that the mass conservation is satisfied. the constants a, b,

and c are defined as.

U C � 'V*+,�- � ./*�-,H � *+,�- � 'V./*�-,. � #1 'V$ sin - cos - . (13)

It exists few models simulating the out-flow of

non-Newtonian fluids in porous medium. While adopting a

law of behavior of Ostwald type [14], an expression of the

dimensionless viscosity function for a non-Newtonian

power-law fluid has been proposed by Pascal. [15, 16] whose

expression, in terms of laminar viscosity function, is,

WX � �Y Z[�E�F\� � [�E�;\�]^_`D (14)

The dimensionless boundary conditions associated with the

no dimensional Eqs. (7) and (8) are


6 ± a� 9 = 0, ��; = 0, ��; = 0, (15)

< = ± )� 9 = 0, ��; = −1, ��; = −1. (16)

Were A=L′/H′. The system is then gorvened by seven

parameters: the aspect ratio of the cavity, A, the modified

thermal Rayleigh number, Ra, the buoyancy ratio, N, the

Lewis number, Le, the anisotropic permeability ratio, K*, the

orientation angle φ and the power-law index, n.

3. Solution

In large aspect ratios (A >> 1), the present problem can be

significantly simplified by the approximation of the parallel

flow in which v=0 and u(x, y)=u(y), in the central part of the

enclosure. Such an approximation follows from the fact that,

for a shallow cavity, the flow in the core of the enclosure is

approximately parallel to the horizontal boundaries. The

temperature and the concentration field, in the central part, can

be divided into the sum of a linear dependence on x and an

unknown function of y. Thus, it is assumed that

9#6, <$ = 9#<$ (17)

�#6, <$ = b� . 6 + c�#<$ (18)

�#6, <$ = b�. 6 + c�#<$ (19)

In the above equations CT and CS are the dimensionless

temperature gradient and the dimensionless concentration

gradient in the x-direction. Substituting Eqs. (17)-(19) into

Eqs. (9)–(11), the governing equations can be reduced to the

following differential equations:

− ddF ZdEdF [dEdF\ef)] = gX.h/jklDm�XeDmL�∗ , (20)

dDnodFD = b� dEdF , (21)

dDn@dFD = pq. b� dEdF . (22)

In Eq. (20), the constant ξ is defined as: ξ=CT + N.CS and

the resulting expressions for the velocity, stream function,

temperature and concentration fields are given by taking into

account the boundary conditions, Eq. (15-16):

3.1. Stream Function

Eq. (20) can be integrated to give the following fully

developed Stream function,

9e#<$ = − e#�XeDmL�∗$`̂ . �gX.h/jklDm�`̂

#eL)$.�[`r`̂\ [#2<$[)L`̂\ − 1\ (23)

3.2. Velocity Distribution

The velocity profile can be gotten, While drifting Eq. (23):

Se#<$ = − F`̂#�XeDmL�∗$`̂ #KC. s/./*�-$`̂

(24)

3.3. Temperature Distribution


developed temperature profile,

�e#6, <$ = b� . 6 − J) te.#F$[`r`̂\#�eL)$ − )

#�$[`r`̂\u b�< − <J) = e.�gX.h/jklDm�`̂

#�XeDmL�∗$`̂ .#eL)$ NOPOQ

(25)

3.4. Concentration Distribution


developed concentration function,

�e#6, <$ = b�. 6 − pqJ) te#F$[`r`̂\#�eL)$ − )

#�$[`r`̂\u b�< − <v (26)

3.5. Determination of CT and CS

The expressions of CT and CS can be deduced by integration

of the following Eqs. (27) and (28), together with the

boundary conditions (15) and (16), by considering the

arbitrary control volume of Figure 1 and connecting with the

region of the parallel flow [4]. This yields:

w [dE^dF �e − ��̂�; \;x y< = 0)/�f)/� , (27)

w [pq. dE^dF �e − ��^�; \;x y< = 0)/�f)/� . (28)

Substituting Eqs. (23)–(26) into Eqs. (27)-(28) and

integrating yields, after some straightforward but laborious

algebra, an expressions of the form:

b� = z̀ �gX.h/jklDm�`̂ .��XeDmL�∗�`̂#�XeDmL�∗$D̂ Lz̀ zD#gX.h/jklDm$D̂ (29)

b� = 8Rz̀ �gX.h/jklDm�`̂ .��XeDmL�∗�`̂#�XeDmL�∗$D̂Lz̀ zD8RD#gX.h/jklDm$D̂ (30)

were:

{) = e#)/�$[`r`̂\�eL) (31)

{� = e#)/�$#`/^$|eL) (32)

Taking into account ξ=CT + NCS and using the Eqs. (29) -

(30) we obtain

�)s}r^^ + ��sDr^^ − �|s ~̂ − ��s `̂ = �#s$ (33)

The constants ε1, ε2, ε3 and ε4 which depend on Ra, K*, φ,

n and Le are given by the following expressions.



�) = #KC././*�-$}̂#pq{){�$�/#�C,�- + '∗$}̂

�� = [ gXjklDm\D̂ {){�#1 + pq�$/#�C,�- + '∗$D̂

�| = [ gXjklDm\~̂ {�{)�#� + pq$/#�C,�- + '∗$~̂

�� = [ gXjklDm\`̂ {)#1 + �pq$/#�C,�- + '∗$`̂ NOOPOOQ

(34)

3.6. Heat Transfer and Mass Transfer

In the case where the cavity is subjected to vertical heat and

mass flux per unit area, we define vertical Nusselt and

Sherwood numbers, evaluated at x=0 by:

�Se = )�̂ # ,f .�$f�̂ # , .�$ (35)

�ℎe = )�^# ,f .�$f�^# , .�$ (36)

By substituting Eqs. (25) and (26) in Eqs. (35) and (36), we

obtain:

�Se = ��XeDmL�∗�`̂#�XeDmL�∗$`̂ f�oz̀ #gX.h/jklDm$`̂ (37)

�ℎe = ��XeDmL�∗�`̂#�XeDmL�∗$`̂ f8R�oz̀ #gX.h/jklDm$`̂ (38)

with the following limits

�� #��^$gX → = 1, �� #M�^$gX → = 1. (39)

4. Results and Discussion

4.1. Case of a Non-newtonian Fluid Saturating Anisotropic

Porous Medium such as (φ=0°)

When the principal axes of the porous matrix (K1, K2)

coincides with the axial (Ox,Oy), we have φ=0°, cos(φ)=1,

tan(φ)=0 and Eq. (23) - (26) Become:

9e#<$ = − e#�∗$`̂ . #gX.h$`̂

#eL)$.�[`r`̂\ [#2<$[)L`̂\ − 1\ (40)

Se#<$ = − F`̂#�∗$`̂ #KC. s$`̂

(41)

�e#6, <$ = b� . 6 − e.#gX.h$`̂#�∗$`̂ .#eL)$ te.#F$[`r`̂\

#�eL)$ − )#�$[`r`̂\u b�< − < (42)

�e#6, <$ = b�. 6 − pq e.#gX.h$`̂#�∗$`̂ .#eL)$ te#F$[`r`̂\

#�eL)$ − )#�$[`r`̂\u b�< − < (43)

4.2. Case of a Non-newtonian Fluid Saturating Isotropic

Porous Medium

When the permeability is the same in all directions (i.e. for

an isotropic porous layer), we have: K1=K2. Which implies

that K*=1. The above result (Eq. 40-43), in the case K*=1,

reduces to

9e#<$ = − e.#gX.h$`̂#eL)$.�[`r`̂\ [#2<$[)L`̂\ − 1\ (44)

Se#<$ = −< `̂#KC. s$`̂ (45)

�e#6, <$ = b� . 6 − e.#gX.h$`̂#eL)$ te.#F$[`r`̂\

#�eL)$ − )#�$[`r`̂\u b�< − <(46)

�e#6, <$ = b� . 6 − pq e.#gX.h$`̂.#eL)$ te#F$[`r`̂\

#�eL)$ − )#�$[`r`̂\u b�< − < (47)

which is reported in the past by Benhadji and Vasseur (2001).

4.3. Case of Newtonian Fluid Saturating Anisotropic Porous

Medium

By introducing into Eq. (23)– (26), n=1, we obtain:

9)#<$ = − �gX.h/jklDm��#�XeDmL�∗$ . [<� − )�\ (48)

S)#<$ = − �gX.h/jklDm�#�XeDmL�∗$ < (49)

�)#6, <$ = b� . 6 − �gX.h/jklDm��#�XeDmL�∗$ [FD| − )�\ b�< − <, (50)

�)#6, <$ = b�. 6 − pq �gX.h/jklDm��#�XeDmL�∗$ [FD| − )�\ b�< − < (51)

4.4. Case of Newtonian Fluid Saturating Isotropic Porous

Medium

When the permeability is the same in all directions (i.e. for

an isotropic porous layer), we have: K1=K2. Which implies

that K*=1. The above result (Eq. 48 - 51), in the case K*=1

and -=0°, reduces to

9)#<$ = − gX.h� . [<� − )�\, (52)

S)#<$ = −KC. s. <, (53)

�)#6, <$ = b� . 6 − gX.h� [FD| − )�\ b�< − <, (54)

�)#6, <$ = b�. 6 − 8R.gX.h� [FD| − )�\ b�< − <. (55)

These results are in agreement with the solutions obtained

in the past by AMARI et al. and KALLA et al. [17, 19].

4.5. Effect of Anisotropy Parameters on Heat Transfer and

Mass Transfer

The determination of the constants CT and CS depends on

the solution of the equation f (ξ)=0. A programming using

the Fortran software is done and allowed us to draw the

curve (figure 2). This curve indicates in case of a

Newtonian fluid (n=1), three solutions namely: ξ=0,

ξ=0.3213 and ξ=0.9337.


Figure 2. Solution of f (ξ)=0.

The effect of varying of Ra, the Rayleigh number and of K*,

the permeaility ratio on the Sherwood number, Sh, is illustraded

in Figure 3 for n=1, ξ=0.3213, Le=4 and φ=0°. It observes that

when the Rayleigh number increases, the Sherwood number for

mass transfer increases too. It is also observed that for low

values of the Rayleigh number, the effect of anisotropy is also

low. In other words, upon increasing Ra, the effect of

anisotropy becomes more and more important. This is

explained by the fact that when the Rayleigh number tends

towards to zero, the sherwood number tends towards unity (Eq.

39). Figure 3 also shows that the anisotropy ratio K* has a great

influence on the Sherwood number. Indeed, the anisotropic

ratio K* decreases with an increase of the Sherwood number,

Sh. Moreover, compared to isotropic situation (K*=1) the

Sherwood number for mass transfer is enhanced when K* < 1,

and decreased when K* > 1. It results then, when the principal

axes of anisotropy are oriented in a direction coinciding with

the coordinate axes (i.e., φ=0°), the Sherwood number for mass

transfer increases (or decreases) when the permeability in the

vertical direction (K1) is smaller (or higher) than the

permeability in the horizontal direction (K2).

Figure 3. The effect of varying of Ra, the Rayleigh number and of K*, the

permeaility ratio on the Sherwood number, Sh.

The effect of the anisotropic angle φ and of the Rayleigh

number, Ra, on the Sherwood number for mass transfer, Sh, is

illustrated in Figure 4 when n=1, ξ=0.3213, Le=4 and K*=0.5.

It is seen that for a given value of the Rayleigh number, the

Sherwood number increases with the decrease of the

anisotropic angle φ when the permeability ratio is made

smaller than 1 (i.e., K∗< 1).

Figure 4. The effect of the anisotropic angle φ and of the Rayleigh number, Ra,

on the Sherwood number for mass transfer, Sh.

Independently of φ and K*, the Sherwood number for mass

transfer increases with the increase of Rayleigh number. For

φ=90°, the Sherwood number is minimal and maximal when

φ=0°. It is evident from the analysis of this result that the

characteristic parameter of the mass transfer (Sh) is minimal

(or maximal) when the main axis having the most elevated

permeability of the porous layer is perpendicular (or parallel)

to the gravity. On the other hand, results illustrated in Figure 5

show that, when the permeability ratio is made superior than 1

(i.e., K* > 1), for φ=0°, the Sherwood number is minimal and

maximal when φ=90°.

Figure 5. Effect of the anisotropic angle φ and of K*, on the Sherwood

number for mass transfer, Sh.

The evolution of the Nussselt number according to the

Rayleigh number and for different values of K* when n=1,

ξ=0.3213, Le=4, and φ=0◦, is illustrated in Figure 6. The results

show that Nusselt number is an increasing function of the

Rayleigh number. Then, compared to isotropic situation (K*=1)



and for a given value of Rayleigh number, the Nussselt number

for heat transfer is enhanced when K* < 1, and decreased when

K* > 1. It results then, when the principal axes of ani-sotropy

are oriented in a direction coinciding with the coordinate axes

(i.e., φ=0°), the enhanced for heat transfer increases (or

decreases) when the permeability in the vertical direction (K1)

is smaller (or higher) than the permeability in the horizontal

direction (K2). Otherwise, results illustrated in Figure 7 show

that, when the permeability ratio is made superior than 1 (i.e.,

K* > 1), for φ=0°, the Nussselt number for heat transfer is

minimal and maximal when φ=90°.

Figure 6. Evolution of the Nussselt number according to the Rayleigh number

and for different values of K*.

Figure 7. Effect of the anisotropic angle φ and of K*, on the Nusselt number.

5. Conclusion

In this investigation, an analytical study of heat and mass

transfer is studied in two-dimensional horizontal shallow

enclosure, filled with non-Newtonian power-law fluids.

Our research concerns the influence of hydrodynamic

anisotropy on the heat and mass transfer. An analytical

model, based on the parallel flow approximation, is

proposed for the case of a shallow layer (A >> 1), to

determine the effect of anisotropic parameters of the porous

matrix on the speed field, the temperature field, the

concentration field and the heat transfer. The main

conclusions of the present study are:

The Sherwood number for mass transfer is an increasing

function of the Rayleigh number.

Compared to isotropic situation (K*=1) the Sherwood

number for mass transfer is enhanced when K* < 1, and

decreased when K* > 1.

The characteristic parameter of the mass transfer (Sh) is

minimal (or maximal) when the main axis having the most

elevated permeability of the porous layer is perpendicular (or

parallel) to the gravity.

The enhanced for heat transfer increases (or decreases) when

the permeability in the vertical direction (K1) is smaller (or

higher) than the permeability in the horizontal direction (K2).

References

[1] Jaluria, Y. (2003). Thermal processing of materials: From basic research to engineering. Journal of Heat Transfer 125, 957-979.

[2] Getachew, D., D. Poulikakos and W. J. Minkowycz. 1998. “Double-diffusive in a porous cavity saturated with non-Newtonian Fluid”. Journal of Thermophysics and Heat Transfer 12, 437-446.

[3] Benhadji, K. and P. Vasseur. 2001. “Double diffusive convection in a shallow porous cavity filled with a non-Newtonian fluid”. International Communications in Heat and Mass Transfer 28, 763-772.

[4] Makayssi, T., M. Lamsaadi, M. Nami, M. Hasnaoui, A. Raji and A. Bahlaoui. 2008. “Natural double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids: the cooperating case”, Energy conversion and Management 49, 2016-2025.

[5] Srinivasacharya, D., Swamy Reddy, G. 2012. Double Diffusive Natural Convection in Power-Law Fluid Saturated Porous Medium with Soret and Dufour Effects J. of the Braz. Soc. of Mech. Sci. Eng. Vol. XXXIV, No. 4, pp. 525-530.

[6] Salma Parvin, Rehena Nasrin, Alim, M. A., Hossain, N. F. 2013 Double diffusive natural convective flow characteristics in a cavity Procedia Engineering Vol. 56, pp. 480 488.

[7] Raju Chowdhury, Salma Parvin, and Md. Abdul Hakim Khan. 2016. Finite element analysis of double-diffusive natural convection in a porous triangular enclosure filled with Al2O3-water nanofluid in presence of heat generation Heliyon. 2016 Aug; 2 (8): e00140.

[8] Ariyan Zare Ghadi, Ali Haghighi Asl, Mohammad Sadegh Valipour. 2014. Numerical modelling of double-diffusive natural convection within an arc shaped enclosure filled with a porous medium Journal of Heat and Mass Transfer Research Vol. 1 pp. 83-91.

[9] Sabyasachi Mondal and Precious Sibanda. 2016. An Unsteady Double-Diffusive Natural Convection in an Inclined Rectangular Enclosure with Different Angles of Magnetic Field International Journal of Computational Methods, Vol. 13, No. 041641015.


[10] BIHICHE, K., LAMSAADI, M., NAMI, M., ELHARFI, H., KADDIRI, M., LOUARAYCHI, A. 2017. Double-diffusive and Soret-induced convection in a shallow horizontal layer lled with non-Newtonian power-law uids 13me Congrs de Mcanique 11 - 14 Avril 2017 (Mekns, MAROC).

[11] Girinath Reddy, M., Dinesh, P. A. 2018. Double Diffusive Convection and Internal Heat Generation with Soret and Dufour Effects over an Accelerating Sur- face with Variable Viscosity and Permeability Advances in Physics Theories and Applications, Vol. 69, pp. 7-25.

[12] Mahapatra, T. R., Saha, B. C. Pal, D. 2018 Magnetohydrodynamic double- diffusive natural convection for nanofluid within a trapezoidal enclosure. Comp. Appl. Math. 37, 61326151 (2018) doi: 10.1007/s40314-018-0676-5.

[13] Abdelraheem M. Aly and Mitsuteru Asai (October 22nd 2015). Double- Diffusive Natural Convection with Cross-Diffusion Effects in an Anisotropic Porous Enclosure Using ISPH Method, Mass Transfer - Advancement in Process Modelling, Marek Solecki, Intech Open, DOI: 10.5772/60879.

[14] Ostwald, W. 1925. “Ueber die Geschwindigkeitsfunktion der Viskositat Disperser Système”, L, Kolloïd-Z, Vol. 36, pp. 99-117.

[15] PASCAL, H. 1983. “Rheological Behaviour Effect of Non Newtonian Fluids on Steady and Unsteady Flow Through a Porous Media”, International Journal for Numerical and Analytical Methods in Geomechanics, vol. 7, pp. 289-303.

[16] PASCAL, H. 1986. “Rheological effects of Non Newtonian Behaviour of Displacing Fluids on Stability of a Moving Interface in Radial Oil Displacement Mechanism in Porous Media”, International Journal Engineering Science, vol. 24, No. 9, pp. 1465-1476.

[17] AMARI, B., VASSEUR, P., and BILGEN, E. 1994. “Natural Convection of Non Newtonian Fluids in a Horizontal Porous Layer”, Wrme-und Stoffbertragung, vol. 29, pp. 185-193.

[18] MAMOU, M., VASSEUR, P., BILGEN, E., and GOBIN, D. 1995. “Double- Diffusive Convection in an Inclined Slot Filled with Porous Medium”, Eur. J. Me- chanics, B/Fluids, vol. 14, pp. 629-652.

[19] KALLA, L., MAMOU M., VASSEUR, P. and ROBILLARD, L. 1999. “Mul- tiple Steady States for Natural Convection in a Shallow Porous Cavity Subject to Uniform Heat Fluxes”, International Communications in Heat Mass Transfer, vol. 26, No. 6, pp. 761-770.

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